## Abstract

Motivated by an error-correcting generalization of Bachet's weights problem, we define and classify relaxed complete partitions. We show that these partitions enjoy

a succinct description in terms of lattice points in polyhedra, with adjustments in the error being commensurate with translations in the defining hyperplanes. Our main result is that the enumeration of the minimal such partitions (those with fewest possible parts) is achieved via Brion's formula. This generalizes work of Park on classifying complete partitions and that of Rodseth on enumerating minimal complete partitions.

a succinct description in terms of lattice points in polyhedra, with adjustments in the error being commensurate with translations in the defining hyperplanes. Our main result is that the enumeration of the minimal such partitions (those with fewest possible parts) is achieved via Brion's formula. This generalizes work of Park on classifying complete partitions and that of Rodseth on enumerating minimal complete partitions.

Original language | English |
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Number of pages | 18 |

Journal | Integers : Electronic Journal of Combinatorial Number Theory |

Volume | 15 |

Issue number | A |

Publication status | E-pub ahead of print - 26 Nov 2015 |

## Keywords

- integer partitions
- bachet's problem
- Brion's formula
- weighing pan
- polytopes
- number theory
- enumerating functions
- geometric number theory